Calculus of Real and Complex Variables

Chapter 11Degree Theory, An Introduction

This chapter is on the Brouwer degree, a very useful concept with numerous and important applications.
The degree can be used to prove some difficult theorems in topology such as the Brouwer fixed point
theorem, the Jordan separation theorem, and the invariance of domain theorem. A couple of these
big theorems have been presented earlier, but when you have degree theory, they get much
easier. Degree theory is also used in bifurcation theory and many other areas in which it is an
essential tool. The degree will be developed for ℝn first. When this is understood, it is not too
difficult to extend to versions of the degree which hold in Banach space. There is more on
degree theory in the book by Deimling [37] and much of the presentation here follows this
reference. Another more recent book which is really good is [42]. This is a whole book on degree
theory.

The original reference for the approach given here, based on analysis, is [60] and dates from 1959. The
degree was developed earlier by Brouwer and others using different methods.

To give you an idea what the degree is about, consider a real valued C1 function defined on an interval,
I, and let y ∈ f

(I)

be such that f′

(x)

≠0 for all x ∈ f−1

(y)

. In this case the degree is the sum of the signs
of f′

(x)

for x ∈ f−1

(y)

, written as d

(f,I,y)

.

PICT

In the above picture, d

(f,I,y)

is 0 because there are two places where the sign is 1 and two where it is
−1.

The amazing thing about this is the number you obtain in this simple manner is a specialization of
something which is defined for continuous functions and which has nothing to do with differentiability. An
outline of the presentation is as follows. First define the degree for smooth functions at regular values and
then extend to arbitrary values and finally to continuous functions. The reason this is possible is an integral
expression for the degree which is insensitive to homotopy. It is very similar to the winding number
presented in the part of the book introducing complex analysis. The difference between the two is that with
the degree, the integral which ties it all together is taken over the open set while the winding number is
taken over the boundary.

In this chapter Ω will refer to a bounded open set.

Definition 11.0.1For Ω a bounded open set, denote by C

(Ω)

the set of functions which are restrictionsof functions in Cc

(ℝn )

to Ωand by Cm

(Ω)

,m ≤∞ the space of restrictions of functions in Ccm

(ℝn)

toΩ. If f ∈ C

(-)
Ω

the symbol f will also be used to denote a function defined on ℝnequalling f on Ωwhenconvenient. The subscript c indicates that the functions have compact support. The norm in C

(-)
Ω

isdefined as follows.

{ --}
||f||∞ ≡ sup |f (x)| : x ∈Ω .

If the functions take values in ℝnwrite Cm

(-- )
Ω;ℝn

or C

(-- )
Ω;ℝn

for these functions if there is nodifferentiability assumed. The norm on C

(-- )
Ω;ℝn

is defined in the same way as above,

{ --}
||f||∞ ≡ sup |f (x)| : x ∈ Ω .

Of course if m = ∞, the notation means that there are infinitely many derivatives. Also, C

(Ω;ℝn)

consistsof functions which are continuous on Ω that have values in ℝnand Cm

Note that, by applying the Tietze extension theorem to the components of the function, one can always
extend a function continuous on Ω to all of ℝn so there is no loss of generality in simply regarding functions
continuous on Ω as restrictions of functions continuous on ℝn. Next is the idea of a regular
value.

Definition 11.0.2For W an open set in ℝnand g ∈ C1

n
(W ;ℝ )

y is called a regular value of g ifwhenever x ∈ g−1

(y)

, det

(Dg (x))

≠0. Note that if g−1

(y)

= ∅, it follows that y is a regular valuefrom this definition. Denote by Sgthe set of singular values of g, those y such that det

(Dg (x))

= 0
for some x ∈ g−1

(y)

.

Also, ∂Ω will often be referred to. It is those points with the property that every open set (or open ball)
containing the point contains points not in Ω and points in Ω. Then the following simple lemma will be
used frequently.

Lemma 11.0.3Define ∂U to be those points x with the property that for every r > 0, B

(x,r)

containspoints of U and points of UC. Then for U an open set,

--
∂U = U ∖ U

Let C be a closed subset of ℝpand let K denote the set of components of ℝp∖C. Then if K is one of thesecomponents, it is open and

∂K ⊆ C

Proof: Let x ∈U∖ U. If B

(x,r)

contains no points of U, then x

∈∕

U. If B

(x,r)

contains no points of
UC, then x ∈ U and so x

∕∈

U∖U. Therefore, U∖U ⊆ ∂U. Now let x ∈ ∂U. If x ∈ U, then since U is open
there is a ball containing x which is contained in U contrary to x ∈ ∂U. Therefore, x

∈∕

U. If x is not a limit
point of U, then some ball containing x contains no points of U contrary to x ∈ ∂U. Therefore, x ∈U∖U
which shows the two sets are equal.

Why is K open for K a component of ℝp∖C? This is obvious because in ℝp an open ball is connected.
Thus if k ∈ K,letting B

(k,r)

⊆ CC, it follows K ∪ B

(k,r)

is connected and contained in CC. Thus
K ∪ B

(k,r)

is connected, contained in CC, and therefore is contained in K because K is maximal with
respect to being connected and contained in CC.

Now for K a component of ℝp∖ C, why is ∂K ⊆ C? Let x ∈ ∂K. If x

∕∈

C, then x ∈ K1, some
component of ℝp∖C. If K1≠K then x cannot be a limit point of K and so it cannot be in ∂K. Therefore,
K = K1 but this also is a contradiction because if x ∈ ∂K then x